This exhilarating contest between two teams in transition was decided by one constant truth: Darren Bent scores goals. Everton should have led by more than a Steven Naismith strike by half-time and Fulham were guilty of haphazard finishing, too, both before and after Dimitar Berbatov's equaliser. So the sides were locked at 1-1 when Bent was introduced in the 66th minute; two minutes later he scored the winning goal, collecting a clever Giorgos Karagounis free-kick at the left-hand side of the box and firing into the net.

Bent had scored against Arsenal on his Fulham debut after joining on loan from Aston Villa in the summer but was able to make only two starts after that before injuring his hamstring. His comeback was well-timed, as Martin Jol was under pressure going into this game due to simmering discontent with his team's tepid start to the season.

Their last victory had come in the previous round of this competition – and even then it took a penalty shootout to overcome Burton Albion – and fans' ire boiled over at the end of their last home match, when a minority called for the manager to be dismissed after a disappointing draw with West Bromwich Albion. Bent eased the Dutchman's burden.

"This was a big relief," said the manager. "It's nice to get back to winning ways."

Jol believes Fulham have struggled so far this season because the likes of Bent, Berbatov and Adel Taraabt have had fitness problems. "We are having to integrate players and help them get fitness at the same time," he said. "Normally we would do that in pre-season but now we have to do it while trying to get results. That has been the problem. But we can only get these talented players if they are not playing well. Bent didn't play for a year [at Villa]. Adel didn't play over summer. Berbatov was out with injury. That has been the problem. We couldn't score enough. We needed players like Bent, who can score out of nothing."

They would have needed a lot more than two if Everton had been ruthless. Roberto Martínez made eight changes to the team that had extended their unbeaten start to the season by winning at West Ham at the weekend but the alterations did not disrupt their they fluency and they immediately forced Fulham on to the back foot. Naismith tested the goalkeeper, David Stockdale, with a soft header in the sixth minute and was given a clearer chance four minutes later but shot straight at David Stockdale after a superb turn and pass by Romelu Lukaku. The Scot was not so forgiving the third time, firing emphatically into the net in the 12th minute after excellent work by Gerard Deulofeu, the 19-year-old loanee from Barcelona, whose jagged dribbling was a delight throughout.

Berbatov could have equalised from the hosts' first real chance but glanced a header wide in the 15th minute. Everton came straight back at them, Sylvain Distin heading against the post moments later. Fulham were looking at risk of a rout.

Fulham came out sharper and stronger in the second half and nearly equalised in the 49th minute, Pajtim Kasami slipping a clever ball through to Alexander Kacaniklic whose shot from 18 yards was pushed behind by Joel Robles. Two minutes later Kacaniklic had an even better opportunity but botched an attempted volley from eight yards. Everton did not heed those warnings and Berbatov brought the sides level in the 54th minute, finishing neatly from 12 yards after a flowing Fulham move.

Brede Hangeland began that move by forcefully dispossessing Lukaku on halfway, a deed that epitomised the home side's improvement. Lukaku was outmuscled again in the 61st minute, allowing Scott Parker to offer Berbatov a chance to inflict further damage, but the Bulgarian failed to find the target from 10 yards. In search of a surer marksman, Jol introduced Bent. The payback was immediate. Seamus Coleman, Deulofeu and Lukaku came close to equalising as Everton besieged the home box late on, but Fulham refused to relinquish the reward for their fightback.